##
**The indirect method: inference based on intermediate statistics – a synthesis and examples.**
*(English)*
Zbl 1100.62025

Summary: This article presents an exposition and synthesis of the theory and some applications of the so-called indirect method of inference. These ideas have been exploited in the field of econometrics, but less so in other fields such as biostatistics and epidemiology. In the indirect method, statistical inference is based on an intermediate statistic, which typically follows an asymptotic normal distribution, but is not necessarily a consistent estimator of the parameter of interest. This intermediate statistic can be a naive estimator based on a convenient but misspecified model, a sample moment or a solution to an estimating equation.

We review a procedure of indirect inference based on the generalized method of moments, which involves adjusting the naive estimator to be consistent and asymptotically normal. The objective function of this procedure is shown to be interpretable as an “indirect likelihood” based on the intermediate statistic. Many properties of the ordinary likelihood function can be extended to this indirect likelihood. This method is often more convenient computationally than maximum likelihood estimation when handling such model complexities as random effects and measurement error, for example, and it can also serve as a basis for robust inference and model selection, with less stringent assumptions on the data generating mechanism. Many familiar estimation techniques can be viewed as examples of this approach.

We describe applications to measurement error, omitted covariates and recurrent events. A dataset concerning prevention of mammary tumors in rats is analyzed using a Poisson regression model with overdispersion. A second dataset from an epidemiological study is analyzed using a logistic regression model with mismeasured covariates. A third dataset of exam scores is used to illustrate robust covariance selection in graphical models.

We review a procedure of indirect inference based on the generalized method of moments, which involves adjusting the naive estimator to be consistent and asymptotically normal. The objective function of this procedure is shown to be interpretable as an “indirect likelihood” based on the intermediate statistic. Many properties of the ordinary likelihood function can be extended to this indirect likelihood. This method is often more convenient computationally than maximum likelihood estimation when handling such model complexities as random effects and measurement error, for example, and it can also serve as a basis for robust inference and model selection, with less stringent assumptions on the data generating mechanism. Many familiar estimation techniques can be viewed as examples of this approach.

We describe applications to measurement error, omitted covariates and recurrent events. A dataset concerning prevention of mammary tumors in rats is analyzed using a Poisson regression model with overdispersion. A second dataset from an epidemiological study is analyzed using a logistic regression model with mismeasured covariates. A third dataset of exam scores is used to illustrate robust covariance selection in graphical models.

### MSC:

62F10 | Point estimation |

62A01 | Foundations and philosophical topics in statistics |

62P10 | Applications of statistics to biology and medical sciences; meta analysis |

### Keywords:

asymptotic normality; bias correction; consistency; efficiency; estimating equations; generalized method of moments; graphical models; indirect inference; indirect likelihood; measurement error; missing data; model selection; naive estimators; omitted covariates; overdispersion; quasi-likelihood; random effects; robustness
PDF
BibTeX
XML
Cite

\textit{W. Jiang} and \textit{B. Turnbull}, Stat. Sci. 19, No. 2, 239--263 (2004; Zbl 1100.62025)

Full Text:
DOI

### References:

[1] | Andersen, P. K., Borgan, Ø., Gill, R. D. and Keiding, N. (1993). Statistical Models Based on Counting Processes. Springer, New York. · Zbl 0769.62061 |

[2] | Berk, R. H. (1966). Limiting behavior of posterior distributions when the model is incorrect. Ann. Math. Statist. 37 51–58. [Correction 37 745–746.] · Zbl 0151.23802 |

[3] | Bickel, P. (1988). Robust estimation. In Encyclopedia of Statistical Sciences (S. Kotz and N. L. Johnson, eds.) 8 157–163. Wiley, New York. |

[4] | Bickel, P. J. and Doksum, K. A. (2001). Mathematical Statistics 1 , 2nd ed. Prentice Hall, Upper Saddle River, NJ. · Zbl 0403.62001 |

[5] | Box, G. E. P. and Tiao, G. C. (1973). Bayesian Inference in Statistical Analysis . Addison–Wesley, London. · Zbl 0271.62044 |

[6] | Breslow, N. (1990). Tests of hypotheses in overdispersed Poisson regression and other quasi-likelihood models. J. Amer. Statist. Assoc. 85 565–571. |

[7] | Broze, L. and Gouriéroux, C. (1998). Pseudo-maximum likelihood method, adjusted pseudo-maximum likelihood method and covariance estimators. J. Econometrics 85 75–98. · Zbl 0962.62015 |

[8] | Carrasco, M. and Florens, J.-P. (2002). Simulation-based method of moments and efficiency. J. Bus. Econom. Statist. 20 482–492. |

[9] | Carroll, R. J., Ruppert, D. and Stefanski, L. A. (1995). Measurement Error in Nonlinear Models . Chapman and Hall, London. · Zbl 0853.62048 |

[10] | Chiang, C. L. (1956). On regular best asymptotically normal estimates. Ann. Math. Statist. 27 336–351. · Zbl 0074.35205 |

[11] | Clark, L. C., Combs, G. F., Turnbull, B. W., Slate, E. H., Chalker, D. K., Chow, J., Davis, L. S., Glover, R. A., Graham, G. F., Gross, E. G., Krongrad, A., Lesher, J. L., Park, H. K., Sanders, B. B., Smith, C. L., Taylor, J. R. and the Nutritional Prevention of Cancer Study Group (1996). Effects of selenium supplementation for cancer prevention in patients with carcinoma of the skin: A randomized controlled trial. J. American Medical Association 276 1957–1963; Editorial 1984–1985. |

[12] | Cox, D. R. (1962). Further results on tests of separate families of hypotheses. J. Roy. Statist. Soc. Ser. B 24 406–424. · Zbl 0131.35801 |

[13] | Cox, D. R. (1972). Regression models and life-tables (with discussion). J. Roy. Statist. Soc. Ser. B 34 187–220. · Zbl 0243.62041 |

[14] | Cox, D. R. (1983). Some remarks on overdispersion. Biometrika 70 269–274. · Zbl 0511.62007 |

[15] | Cox, D. R. and Wermuth, N. (1990). An approximation to maximum likelihood estimates in reduced models. Biometrika 77 747–761. · Zbl 0709.62050 |

[16] | Crowder, M. (1985). Gaussian estimation for correlated binomial data. J. Roy. Statist. Soc. Ser. B 47 229–237. |

[17] | Crowder, M. (2001). On repeated measures analysis with misspecified covariance structure. J. R. Stat. Soc. Ser. B Stat. Methodol. 63 55–62. · Zbl 0976.62068 |

[18] | Dawid, A. P. (1998). Conditional independence. In Encyclopedia of Statistical Sciences , Update Volume (S. Kotz, C. B. Read and D. L. Banks, eds.) 2 146–155. Wiley, New York. |

[19] | de Luna, X. and Genton, M. G. (2001). Robust simulation-based estimation of ARMA models. J. Comput. Graph. Statist. 10 370–387. · Zbl 04567028 |

[20] | de Luna, X. and Genton, M. G. (2002). Simulation-based inference for simultaneous processes on regular lattices. Stat. Comput. 12 125–134. |

[21] | Dempster, A. P., Laird, N. M. and Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm (with discussion). J. Roy. Statist. Soc. Ser. B 39 1–38. · Zbl 0364.62022 |

[22] | Draper, D. (1995). Assessment and propagation of model uncertainty (with discussion). J. Roy. Statist. Soc. Ser. B 57 45–97. · Zbl 0812.62001 |

[23] | Ferguson, T. S. (1958). A method of generating best asymptotically normal estimates with application to the estimation of bacterial densities. Ann. Math. Statist. 29 1046–1062. · Zbl 0089.15402 |

[24] | Fisher, R. A. (1946). Statistical Methods for Research Workers , 10th ed. Oliver and Boyd, Edinburgh. · JFM 64.0544.03 |

[25] | Florey, C. du V., Melia, R. J. W., Chinn, S., Goldstein, B. D., Brooks, A. G. F., John, H. H., Craighead, E. B. and Webster, X. (1979). The relation between respiratory illness in primary schoolchildren and the use of gas for cooking, III—Nitrogen dioxide, respiratory illness and lung function. International J. Epidemiology 8 347–353. |

[26] | Foutz, R. V. and Srivastava, R. C. (1977). The performance of the likelihood ratio test when the model is incorrect. Ann. Statist. 5 1183–1194. · Zbl 0391.62004 |

[27] | Fuller, W. A. (1987). Measurement Error Models . Wiley, New York. · Zbl 0800.62413 |

[28] | Gail, M. H., Santner, T. and Brown, C. C. (1980). An analysis of comparative carcinogenesis experiments based on multiple times to tumor. Biometrics 36 255–266. · Zbl 0463.62098 |

[29] | Gail, M. H., Wieand, S. and Piantadosi, S. (1984). Biased estimates of treatment effect in randomized experiments with nonlinear regressions and omitted covariates. Biometrika 71 431–444. · Zbl 0565.62094 |

[30] | Gallant, A. R. and Long, J. R. (1997). Estimating stochastic differential equations efficiently by minimum chi-squared. Biometrika 84 125–141. · Zbl 0953.62084 |

[31] | Gallant, A. R. and Tauchen, G. (1996). Which moments to match? Econometric Theory 12 657–681. |

[32] | Gallant, A. R. and Tauchen, G. (1999). The relative efficiency of method of moments estimators. J. Econometrics 92 149–172. · Zbl 0956.62030 |

[33] | Genton, M. G. and de Luna, X. (2000). Robust simulation-based estimation. Statist. Probab. Lett. 48 253–259. · Zbl 1122.62308 |

[34] | Genton, M. G. and Ronchetti, E. (2003). Robust indirect inference. J. Amer. Statist. Assoc. 98 67–76. · Zbl 1047.62025 |

[35] | Gouriéroux, C. and Monfort, A. (1993). Simulation-based inference—A survey with special reference to panel-data models. J. Econometrics 59 5–33. · Zbl 0778.62104 |

[36] | Gouriéroux, C., Monfort, A. and Renault, E. (1993). Indirect inference. J. Applied Econometrics 8S 85–118. · Zbl 1448.62202 |

[37] | Hájek, J. (1970). A characterization of limiting distributions of regular estimates. Z. Wahrsch. Verw. Gebiete 14 323–330. · Zbl 0193.18001 |

[38] | Hampel, F. R. (1968). Contributions to the theory of robust estimation. Ph.D. dissertation, Univ. California, Berkeley. |

[39] | Hampel, F. R., Ronchetti, E. M., Rousseeuw, P. J. and Stahel, W. A. (1986). Robust Statistics. The Approach Based on Influence Functions . Wiley, New York. · Zbl 0593.62027 |

[40] | Hand, D. and Crowder, M. (1996). Practical Longitudinal Data Analysis. Chapman and Hall/CRC, London. · Zbl 0885.62002 |

[41] | Hansen, L. P. (1982). Large sample properties of generalized method of moments estimators. Econometrica 50 1029–1054. · Zbl 0502.62098 |

[42] | Hausman, J. A. (1978). Specification tests in econometrics. Econometrica 46 1251–1271. · Zbl 0397.62043 |

[43] | Heckman, J. (1976). The common structure of statistical models of truncation, sample selection and limited dependent variables and a simple estimator for such models. Annals of Economic and Social Measurement 5 475–492. |

[44] | Hengartner, N. W. and Sperlich, S. (2002). Rate optimal estimation with the integration method in the presence of many covariates. Working Paper 01-69, Carlos III de Madrid. Available at http://halweb.uc3m.es/esp/Personal/personas/stefan/papers/ may2002.pdf. |

[45] | Huber, P. J. (1964). Robust estimation of a location parameter. Ann. Math. Statist. 35 73–101. · Zbl 0136.39805 |

[46] | Huber, P. J. (1967). The behavior of maximum likelihood estimates under nonstandard conditions. Proc. Fifth Berkeley Symp. Math. Statist. Probab. 1 221–233. Univ. California Press. · Zbl 0212.21504 |

[47] | Imbens, G. W. (2002). Generalized method of moments and empirical likelihood. J. Bus. Econom. Statist. 20 493–506. |

[48] | Jiang, W. (1996). Aspects of misspecification in statistical models: Applications to latent variables, measurement error, random effects, omitted covariates and incomplete data. Ph.D. dissertation, Cornell Univ. |

[49] | Jiang, W. and Turnbull, B. W. (2003). The indirect method—Robust inference based on intermediate statistics. Technical Report 1377, School of Operations Research and Industrial Engineering, Cornell Univ. Available at http://www.orie.cornell.edu/trlist/trlist.html. |

[50] | Jiang, W., Turnbull, B. W. and Clark, L. C. (1999). Semiparametric regression models for repeated events with random effects and measurement error. J. Amer. Statist. Assoc. 94 111–124. · Zbl 0997.62067 |

[51] | Kent, J. T. (1982). Robust properties of likelihood ratio tests. Biometrika 69 19–27. · Zbl 0485.62031 |

[52] | Kuk, A. Y. C. (1995). Asymptotically unbiased estimation in generalised linear models with random effects. J. Roy. Statist. Soc. Ser. B 57 395–407. · Zbl 0813.62064 |

[53] | Lawless, J. F. and Nadeau, C. (1995). Some simple robust methods for the analysis of recurrent events. Technometrics 37 158–168. · Zbl 0822.62085 |

[54] | Leaderer, B. P., Zagraniski, R. T., Berwick, M. and Stolwijk, J. A. J. (1986). Assessment of exposure to indoor air contaminants from combustion sources: Methodology and application. American J. Epidemiology 124 275–289. |

[55] | Le Cam, L. (1956). On the asymptotic theory of estimation and testing hypotheses. Proc. Third Berkeley Symp. Math. Statist. Probab. 1 129–156. Univ. California Press. · Zbl 0074.13504 |

[56] | Lehmann, E. L. and Casella, G. (1998). Theory of Point Estimation , 2nd ed. Springer, New York. · Zbl 0916.62017 |

[57] | Liang, K. Y. and Zeger, S. L. (1986). Longitudinal data analysis using generalized linear models. Biometrika 73 13–22. · Zbl 0595.62110 |

[58] | Little, R. J. A. (1994). A class of pattern-mixture models for normal incomplete data. Biometrika 81 471–483. · Zbl 0816.62023 |

[59] | MacKinnon, J. G. and Smith, A. A. (1998). Approximate bias correction in econometrics. J. Econometrics 85 205–230. · Zbl 0961.62111 |

[60] | Mammen, E., Linton, O. and Nielsen, J. (1999). The existence and asymptotic properties of a backfitting projection algorithm under weak conditions. Ann. Statist. 27 1443–1490. · Zbl 0986.62028 |

[61] | Mardia, K. V., Kent, J. T. and Bibby, J. (1979). Multivariate Analysis . Academic Press, New York. · Zbl 0432.62029 |

[62] | Mátyás, L., ed. (1999). Generalized Method of Moments Estimation . Cambridge Univ. Press. |

[63] | McCullagh, P. and Nelder, J. A. (1989). Generalized Linear Models , 2nd ed. Chapman and Hall, New York. · Zbl 0588.62104 |

[64] | McFadden, D. (1989). A method of simulated moments for estimation of discrete response models without numerical integration. Econometrica 57 995–1026. · Zbl 0679.62101 |

[65] | Newey, W. K. (1994). Kernel estimation of partial means and a general variance estimator. Econometric Theory 10 233–253. |

[66] | Newey, W. K. and McFadden, D. (1994). Large sample estimation and hypothesis testing. In Handbook of Econometrics (R. F. Engle and D. L. McFadden, eds.) 4 2111–2245. North-Holland, Amsterdam. |

[67] | Pakes, A. and Pollard, D. (1989). Simulation and the asymptotics of optimization estimators. Econometrica 57 1027–1057. · Zbl 0698.62031 |

[68] | Qu, A., Lindsay, B. G. and Li, B. (2000). Improving generalised estimating equations using quadratic inference functions. Biometrika 87 823–836. · Zbl 1028.62045 |

[69] | Quandt, R. E. and Ramsey, J. B. (1978). Estimating mixtures of normal distributions and switching regressions (with discussion). J. Amer. Statist. Assoc. 73 730–752. · Zbl 0401.62024 |

[70] | Rao, C. R. (1973). Linear Statistical Inference and Its Applications , 2nd ed. Wiley, New York. · Zbl 0256.62002 |

[71] | Reid, N. (1988). Influence functions. In Encyclopedia of Statistical Sciences (S. Kotz and N. L. Johnson, eds.) 4 117–119. Wiley, New York. |

[72] | Ronchetti, E. and Trojani, F. (2001). Robust inference with GMM estimators. J. Econometrics 101 37–69. · Zbl 0996.62026 |

[73] | Rosner, B., Spiegelman, D. and Willett, W. C. (1990). Correction of logistic regression relative risk estimates and confidence intervals for measurement error: The case of multiple covariates measured with error. American J. Epidemiology 132 734–745. |

[74] | Rotnitzky, A. and Wypij, D. (1994). A note on the bias of estimators with missing data. Biometrics 50 1163–1170. |

[75] | Schmidt, P. (1982). An improved version of the Quandt–Ramsey MGF estimator for mixtures of normal distributions and switching regressions. Econometrica 50 501–516. · Zbl 0504.62046 |

[76] | Schwarz, G. (1978). Estimating the dimension of a model. Ann. Statist. 6 461–464. JSTOR: · Zbl 0379.62005 |

[77] | Sen, P. K. and Singer, J. M. (1993). Large Sample Methods in Statistics . Chapman and Hall, New York. · Zbl 0867.62003 |

[78] | Sen, S. (1998). Confidence intervals for gene location: The effect of model misspecification and smoothing. Ph.D. dissertation, Dept. Statistics, Univ. Chicago. |

[79] | Serfling, R. J. (1980). Approximation Theorems of Mathematical Statistics . Wiley, New York. · Zbl 0538.62002 |

[80] | Taylor, J. R. (1997). An Introduction to Error Analysis , 2nd ed. University Science Books, Sausalito, CA. |

[81] | Thompson, H. F., Grubbs, C. J., Moon, R. C. and Sporn, M. B. (1978). Continual requirement of retinoid for maintenance of mammary cancer inhibition. Proc. Annual Meeting of the American Association for Cancer Research 19 74. |

[82] | Turnbull, B. W., Jiang, W. and Clark, L. C. (1997). Regression models for recurrent event data: Parametric random effects models with measurement error. Statistics in Medicine 16 853–864. |

[83] | Wedderburn, R. W. M. (1974). Quasi-likelihood functions, generalized linear models and the Gauss–Newton method. Biometrika 61 439–447. · Zbl 0292.62050 |

[84] | Wei, L. J., Lin, D. Y. and Weissfeld, L. (1989). Regression analysis of multivariate incomplete failure time data by modeling marginal distributions. J. Amer. Statist. Assoc. 84 1065–1073. |

[85] | White, H. (1994). Estimation , Inference and Specification Analysis . Cambridge Univ. Press. · Zbl 0860.62100 |

[86] | Whittaker, J. (1990). Graphical Models in Applied Multivariate Statistics . Wiley, New York. · Zbl 0732.62056 |

[87] | Whittemore, A. S. and Keller, J. B. (1988). Approximations for regression with covariate measurement error. J. Amer. Statist. Assoc. 83 1057–1066. · Zbl 0672.62078 |

[88] | Whittle, P. (1961). Gaussian estimation in stationary time series. Bull. Internat. Statist. Inst. 39 105–129. · Zbl 0116.11403 |

[89] | Wright, F. A. and Kong, A. (1997). Linkage mapping in experimental crosses: The robustness of single-gene models. Genetics 146 417–425. |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.